Supplementary variable method for thermodynamically consistent partial differential equations

Abstract

We present a paradigm for developing thermodynamical consistent numerical algorithms for thermodynamically consistent partial differential equation (TCPDE) systems, called the supplementary variable method (SVM). We add a proper number of supplementary variables to the TCPDE system coupled with its energy equation and other deduced equations through perturbations to arrive at a consistent, well-determined, solvable and structurally stable system. The extended system not only reduces to the TCPDE system at specific values of the supplementary variables, but also allows one to retain consistency and solvability after a consistent numerical approximation. Among virtually infinite many possibilities to add the supplementary variables, we present two that maintain thermodynamical consistency in the extended system before and after the approximation. A pseudo-spectral method is used in space to arrive at fully discrete schemes. The new schemes are compared with the energy stable SAV scheme and the fully implicit Crank–Nicolson scheme. The numerical results favor the new schemes in the overall performance.